Which Values Of $x$ Are Solutions To The Equation Below? Check All That Apply. 10 X 2 − 56 = 88 − 6 X 2 10x^2 - 56 = 88 - 6x^2 10 X 2 − 56 = 88 − 6 X 2 A. X = − 3 X = -3 X = − 3 B. X = − 12 X = -12 X = − 12 C. X = 3 X = 3 X = 3 D. X = 3 X = \sqrt{3} X = 3 ​ E. X = − 3 X = -\sqrt{3} X = − 3 ​ F.

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Understanding the Equation

The given equation is a quadratic equation in the form of $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants. In this case, the equation is $10x^2 - 56 = 88 - 6x^2$. To solve for the values of $x$, we need to isolate the variable $x$.

Rearranging the Equation

First, let's rearrange the equation to get all the terms on one side. We can do this by subtracting $88$ from both sides and adding $6x^2$ to both sides:

$$10x^2 - 6x^2 - 56 = 88 - 88$$

This simplifies to:

$$4x^2 - 56 = 0$$

Solving for $x$

Now, we can solve for $x$ by isolating the variable. First, let's add $56$ to both sides to get:

$$4x^2 = 56$$

Next, we can divide both sides by $4$ to get:

$$x^2 = 14$$

Finding the Solutions

To find the solutions for $x$, we need to take the square root of both sides. This gives us:

$$x = \pm \sqrt{14}$$

Checking the Solutions

Now, let's check which of the given solutions match our solutions. We have:

• $x = -3$
• $x = -12$
• $x = 3$
• $x = \sqrt{3}$
• $x = -\sqrt{3}$
• $x = \sqrt{14}$

Analyzing the Solutions

Looking at our solutions, we can see that none of the given solutions match our solutions exactly. However, we can see that $\sqrt{14}$ is close to $\sqrt{3}$, but not exactly the same.

Conclusion

In conclusion, the solutions to the equation $10x^2 - 56 = 88 - 6x^2$ are $x = \pm \sqrt{14}$. None of the given solutions match our solutions exactly, but $\sqrt{14}$ is close to $\sqrt{3}$.

Final Answer

The final answer is:

• $x = \pm \sqrt{14}$

Discussion

This problem is a great example of how to solve quadratic equations and how to check solutions. It's also a good reminder that not all solutions may be exact matches to the given solutions.

Additional Tips

• When solving quadratic equations, make sure to isolate the variable $x$.
• When checking solutions, make sure to check all possible solutions, not just the ones that seem obvious.
• When working with square roots, make sure to simplify the expression before solving for $x$.

Common Mistakes

• Not isolating the variable $x$ when solving quadratic equations.
• Not checking all possible solutions when checking solutions.
• Not simplifying the expression before solving for $x$ when working with square roots.

Real-World Applications

Quadratic equations have many real-world applications, such as:

• Modeling the motion of objects under the influence of gravity.
• Finding the maximum or minimum value of a function.
• Solving problems in physics, engineering, and economics.

Conclusion

In conclusion, solving quadratic equations is an important skill to have in mathematics. By following the steps outlined in this article, you can solve quadratic equations and check solutions with ease. Remember to always isolate the variable $x$, check all possible solutions, and simplify the expression before solving for $x$.

Understanding Quadratic Equations

Quadratic equations are a type of polynomial equation that can be written in the form of $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants. Quadratic equations can be solved using various methods, including factoring, the quadratic formula, and graphing.

Q: What is the quadratic formula?

A: The quadratic formula is a method for solving quadratic equations that is given by:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

Q: How do I use the quadratic formula?

A: To use the quadratic formula, you need to identify the values of $a$, $b$, and $c$ in the quadratic equation. Then, plug these values into the quadratic formula and simplify the expression.

Q: What is the difference between a quadratic equation and a linear equation?

A: A quadratic equation is a polynomial equation of degree 2, while a linear equation is a polynomial equation of degree 1. In other words, a quadratic equation has a squared variable, while a linear equation does not.

Q: Can I solve a quadratic equation by factoring?

A: Yes, you can solve a quadratic equation by factoring if the equation can be written in the form of $(x - r)(x - s) = 0$, where $r$ and $s$ are constants.

Q: What is the relationship between the solutions of a quadratic equation and the graph of the equation?

A: The solutions of a quadratic equation are the x-coordinates of the points where the graph of the equation intersects the x-axis.

Q: Can I use a calculator to solve a quadratic equation?

A: Yes, you can use a calculator to solve a quadratic equation. Most calculators have a built-in quadratic formula function that you can use to solve the equation.

Q: What is the significance of the discriminant in a quadratic equation?

A: The discriminant is the expression $b^2 - 4ac$ in the quadratic formula. If the discriminant is positive, the equation has two real solutions. If the discriminant is zero, the equation has one real solution. If the discriminant is negative, the equation has no real solutions.

Q: Can I solve a quadratic equation with complex solutions?

A: Yes, you can solve a quadratic equation with complex solutions using the quadratic formula. The solutions will be in the form of $x = a \pm bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit.

Q: What is the relationship between the solutions of a quadratic equation and the graph of the equation in the complex plane?

A: The solutions of a quadratic equation are the points in the complex plane where the graph of the equation intersects the complex axis.

Q: Can I use a computer program to solve a quadratic equation?

A: Yes, you can use a computer program to solve a quadratic equation. Most computer algebra systems, such as Mathematica or Maple, have built-in functions for solving quadratic equations.

Q: What is the significance of the quadratic equation in real-world applications?

A: The quadratic equation has many real-world applications, such as modeling the motion of objects under the influence of gravity, finding the maximum or minimum value of a function, and solving problems in physics, engineering, and economics.

Conclusion

In conclusion, quadratic equations are an important topic in mathematics that have many real-world applications. By understanding the quadratic formula, factoring, and graphing, you can solve quadratic equations and apply them to various problems in science and engineering.